M2-1-2: Introduction to Network Analysis

Some housekeeping (again), installing necessary packages.

list.of.packages <- c("igraph", "tidygraph", "ggraph")
new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) install.packages(new.packages)
rm(list.of.packages, new.packages)

Introduction

Networks… So what?

So, before we talk about networks, one thing upfront… why should we? I mean, they undeniably look pretty, don’t they?

Somehow, the visualization of networks fascinates the human mind (find a short TED talk on networks and how they depict our world here), and has even inspired an own art movement, networkism (see some examples here).

Yet, besides that, is there an analytical value for a data scientist to bother about networks?

Networks in R

There are a number of applications designed for network analysis and the creation of network graphs such as gephi and cytoscape. Though not specifically designed for it, R has developed into a powerful tool for network analysis.

Significant network analysis packages for R include the network, sna, and igraph package. In addition, Thomas Lin Pedersen has recently released the tidygraph package that leverage the power of igraph in a manner consistent with the tidyverse workflow. Even better, he tops it up with ggraph, a consistent ´ggplot2´-look-and-feel network visualization package.

R can also be used to make interactive network graphs with the htmlwidgets framework that translates R code to JavaScript. Cool implementations thereof are the vizNetwork and networkD3 packages.

As analytical tool, I will in this lab mostly use igraph. In terms of functions, it is pretty much equivalent to network, yet slightly more powerful, better integrated, and maintained. Since both packages have many of the same functions, better don’t load them both at once.

The Basic Structure of Networks

The Basic Jargon

First of all, what is a network? Plainly speaking, a network is a system of elements which are connected by some relationship. The vocabulary can be a bit technical and even inconsistent between different disciplines, packages, and software. The whole system is (surprise, surprise) usually called a network or graph. The elements are commonly referred to as nodes (system theory jargon) or vertices (graph theory jargon) of a graph, while the connections are edges or links. I will mostly refer to the elements as nodes, and their connections as edges.

Generally, networks are a form of representing relational data. This is a very general tool that can be applied to many different types of relationships between all kind of elements. The content, meaning, and interpretation for sure depends on what elements we display, and which types of relationships. For example:

• In Social Network Analysis:
  • Nodes represent actors (which can be persons, firms and other socially constructed entities)
  • Edges represent relationships between this actors (friendship, interaction, co-affiliation, similarity ect.)
• Other types of network
  • Chemistry: Interaction between molecules
  • Computer Science: The wirld-wide-web, inter- and intranet topologies
  • Biology: Food-web, ant-hives

The possibilities to depict relational data are manifold. For example:

• Relations among persons
  • Kinship: mother of, wife of…
  • Other role based: boss of, supervisor of…
  • Cognitive/perceptual: knows, aware of what they know…
  • Affective: likes, trusts…
  • Interaction: give advice, talks to…
  • Affiliation: belong to same clubs, shares same interests…
• Relations among organizations
  • As corporate entities
  • Buy from / sell to, leases to, outsources to
  • Owns shares of, subsidiary of
  • Joint ventures, strategic alliances
  • Via their members
    • Personnel flows
    • Interlocking directorates
    • Personal friendships
    • Co-memberships
• Relations other (non-social) entities
  • Patents
    • Patents citing other patents
    • Co-occurrence of technological classes *Research fields
    • through citations
    • through people co-affiliated with fields *Sectors
    • input-output relations
    • Labor mobility *Technologies
    • Patent IPC classes
    • Semantic co-occurrence

Note: Content matters! Each relation yields a different structure & has different effects. Theories might make sense on inter-personal, but not inter-organizational or non-social context.

The Data-Structure of Relational Data

Edgelist

MOst real world relational data is to be found in what we call an edge list, a dataframe that contains a minimum of two columns, one column of nodes that are the source of a connection and another column of nodes that are the target of the connection. The nodes in the data are identified by unique IDs. If the distinction between source and target is meaningful, the network is directed. If the distinction is not meaningful, the network is undirected (more on that later). So, every row that contains the ID of one element in column 1, and the ID of another element in column 2 indicates that a connection between them exists. An edge list can also contain additional columns that describe attributes of the edges such as a magnitude aspect for an edge. If the edges have a magnitude attribute the graph is considered weighted (more on that later). Below an example ofa minimal edge list created with the tibble() function.

edge_list <- tibble(from = c(1, 2, 2, 3, 4), to = c(2, 3, 4, 2, 1))
edge_list

Sometimes it is preferable to also create a separate node list. At its simplest, a node list is a data frame with a single column - which I will label as “id” - that lists the node IDs found in the edge list. The advantage of creating a separate node list is the ability to add attribute columns to the data frame such as the names of the nodes or any kind of groupings.

library(tidyverse)
node_list <- tibble(id = 1:4, group = sample(letters[1:2], 4, replace = TRUE))
node_list

Adjacency Matrix

A second popular form of network representation is the adjacency-matrix (also called socio-matrix). It is represented as a \(n*n\) matrix, where \(n\) stands for the number of elements of which their relationships should be represented. The value in the cell that intercepts row \(n\) and column \(m\) indicates if an edge is present (=1) or absent (=0).

Tip: Given an edgelist, an adjacency matrix can easily be produced by crosstabulating:

adj_matrix <- table(edge_list) %>% as.matrix()
adj_matrix

    to
from 1 2 3 4
   1 0 1 0 0
   2 0 0 1 1
   3 0 1 0 0
   4 1 0 0 0

Generating a Graph Object in igraph

To create an igraph object from an edge-list data frame we can use the graph_from_data_frame() function, which is a bit more straight forward than network(). There are three arguments in the graph_from_data_frame() function: d, vertices, and directed. Here, d refers to the edge list, vertices to the node list, and directed can be either TRUE or FALSE depending on whether the data is directed or undirected. By default, graph.data.frame() treats the first two columns of the edge list and any remaining columns as edge attributes.

library(igraph)
g <- graph_from_data_frame(d = edge_list, vertices = node_list, directed = FALSE)
g

IGRAPH d85cc1e UN-- 4 5 -- 
+ attr: name (v/c), group (v/c)
+ edges from d85cc1e (vertex names):
[1] 1--2 2--3 2--4 2--3 1--4

Lets inspect the resulting object. An igraph graph object summary reveals some interesting informations.

• First, it tells us the graph-type: undirected UN, or directed DN
• Afterwards, the number of nodes (4), and edges (6)
• Followed by the node attributes (node level variables), which in this case are only their name (attr: name (v/c))
• Lastly, a list of all existing edges. Note: n--m indicates an undirected, n->m an directed edge.

Lets take a look at the structure of the object:

glimpse(g[[1]])

List of 1
 $ 1: 'igraph.vs' Named int [1:2] 2 4
  ..- attr(*, "names")= chr [1:2] "2" "4"
  ..- attr(*, "env")=<weakref> 
  ..- attr(*, "graph")= chr "d85cc1e0-c674-11e8-816a-b7a131578cf9"

We see, the object has a list-format, consisting of sepperate lists for every node, containing some attributes which are irrelevant now, and an edgelist for every node, capturing its ego-network (eg., .. ..- attr(*, "names")= chr [1:2] "2" "4")

We can also plot it to take a look. igraph object can be directly used with the plot() function. The results can be adjusted with a set of parameters we will discover later. It’s not super pretty, therefore we will later also explore more powerfull plotting tools for rgaphs. However, its quick&dirty, so lets take it like that for now.

plot(g)

Yeah, that’s the graph. We We can also use the adjacency matrix to create the same graph.

g <- graph_from_adjacency_matrix(adj_matrix, mode = "undirected")
g

IGRAPH fcf6ff5 UN-- 4 4 -- 
+ attr: name (v/c)
+ edges from fcf6ff5 (vertex names):
[1] 1--2 1--4 2--3 2--4

We can inspect and manipulate the nodes via V(g) (V for vertices, its graph-theory slang), and edges with E(g)

V(g)

+ 4/4 vertices, named, from fcf6ff5:
[1] 1 2 3 4

E(g)

+ 4/4 edges from fcf6ff5 (vertex names):
[1] 1--2 1--4 2--3 2--4

We can also use most of the base-R slicing&dicing.

V(g)[1:3]

+ 3/4 vertices, named, from fcf6ff5:
[1] 1 2 3

E(g)[2:4]

+ 3/4 edges from fcf6ff5 (vertex names):
[1] 1--4 2--3 2--4

Remember, it’s a list-object. So, if we just want to have the values, we have to use the double bracket [[x]].

V(g)[[1:3]]

+ 3/4 vertices, named, from fcf6ff5:

We can also use the $ notation.

V(g)$name

[1] "1" "2" "3" "4"

Networks are coming…

Build the graph

rm(list=ls())
files <- list.files(path ="../data/GoT/", full.names = TRUE)
files

 [1] "../data/GoT/asoiaf-all-edges.csv"    "../data/GoT/asoiaf-all-nodes.csv"   
 [3] "../data/GoT/asoiaf-book1-edges.csv"  "../data/GoT/asoiaf-book1-nodes.csv" 
 [5] "../data/GoT/asoiaf-book2-edges.csv"  "../data/GoT/asoiaf-book2-nodes.csv" 
 [7] "../data/GoT/asoiaf-book3-edges.csv"  "../data/GoT/asoiaf-book3-nodes.csv" 
 [9] "../data/GoT/asoiaf-book4-edges.csv"  "../data/GoT/asoiaf-book4-nodes.csv" 
[11] "../data/GoT/asoiaf-book45-edges.csv" "../data/GoT/asoiaf-book45-nodes.csv"
[13] "../data/GoT/asoiaf-book5-edges.csv"  "../data/GoT/asoiaf-book5-nodes.csv" 
[15] "../data/GoT/union_characters.RDS"    "../data/GoT/union_edges.RDS"        

edges.cooc.all <- fread(files[1], data.table = FALSE) 
head(edges.cooc.all)

So, that’s what we have, a classical edgelist, with id1 in column 1 and id2 in column2. Note, the edges are in this case weighted.I don’t like the sepperating “-” between in the names, lets get rid of them.

colnames(edges.cooc.all) <- tolower(colnames(edges.cooc.all))
edges.cooc.all %<>%
  mutate(source = gsub("-", " ", source),
         target = gsub("-", " ", target)) 

Ok, lets see how many characters we have overal.

edges.cooc.all %>%
  select(-type) %>%
  gather(x, name, source:target) %>% 
  n_distinct(.$name)

[1] 5646

chars.main <- edges.cooc.all %>%
  select(-type) %>%
  gather(x, name, source:target) %>%
  group_by(name) %>%
  summarise(sum_weight = sum(weight)) %>%
  ungroup() %>%
  arrange(desc(sum_weight)) %>%
  top_n(50)
head(chars.main)

So far so good, if we only go by edge weights, Tyrion is going to make it…. my favorite anyhow…

However, lets reduce our edgelist to this main characters, just to warm up and keep the overview.

edges.cooc <- edges.cooc.all %>%
  filter(source %in% chars.main$name & target %in% chars.main$name) %>%
  select(source, target, weight)

g <- graph_from_data_frame(d = edges.cooc, directed = FALSE)
g

IGRAPH 10c1de6 UNW- 50 402 -- 
+ attr: name (v/c), weight (e/n)
+ edges from 10c1de6 (vertex names):
 [1] Aemon Targaryen (Maester Aemon)--Grenn             Aemon Targaryen (Maester Aemon)--Jeor Mormont     
 [3] Aemon Targaryen (Maester Aemon)--Jon Snow          Aemon Targaryen (Maester Aemon)--Mance Rayder     
 [5] Aemon Targaryen (Maester Aemon)--Robert Baratheon  Aemon Targaryen (Maester Aemon)--Samwell Tarly    
 [7] Aemon Targaryen (Maester Aemon)--Stannis Baratheon Arya Stark                     --Bran Stark       
 [9] Arya Stark                     --Catelyn Stark     Arya Stark                     --Cersei Lannister 
[11] Arya Stark                     --Eddard Stark      Arya Stark                     --Gregor Clegane   
[13] Arya Stark                     --Ilyn Payne        Arya Stark                     --Jaime Lannister  
[15] Arya Stark                     --Joffrey Baratheon Arya Stark                     --Jon Snow         
+ ... omitted several edges

Note that this co-occurence network is weighted (number of co-occurence), and undirected

is_weighted(g)

[1] TRUE

is_directed(g)

[1] FALSE

Inspect the graph

Overal graph attributes

We already know from the summary, but we can also count the number of nodes and edges as follows:

# Count number of edges
gsize(g)

[1] 402

# Count number of vertices
gorder(g)

[1] 50

We can give the graph a first plot to see what happens there. It’s not pretty, but we will fine-tune it later

plot(g)

We already see that some nodes are not connected (isolated), so lets drop them for our network analysis.

g <- delete_edges(g, E(g)[weight < 20])
g <- delete_vertices(g, degree(g) == 0)

Edges

# Find all edges that include "Britt"
E(g)[[inc('Daenerys Targaryen')]]  

+ 6/189 edges from 17b0991 (vertex names):

# Find all pairs that spend 4 or more hours together per week
E(g)[[weight >= 150]]

+ 11/189 edges from 17b0991 (vertex names):

hist(E(g)$weight)

Nodes

lets see who is the most central figure in this network of interactions

Degree centrality

degree(g) 

Aemon Targaryen (Maester Aemon)                      Arya Stark                 Barristan Selmy 
                              3                               9                               5 
                     Bran Stark                Brienne of Tarth                           Bronn 
                             14                               4                               2 
                  Catelyn Stark                Cersei Lannister              Daenerys Targaryen 
                             16                              20                               6 
                 Davos Seaworth                           Drogo                    Eddard Stark 
                              2                               2                              18 
                   Edmure Tully                  Gregor Clegane                           Grenn 
                              3                               4                               2 
               Hizdahr zo Loraq                           Hodor                      Ilyn Payne 
                              2                               4                               4 
                Jaime Lannister                    Jeor Mormont               Joffrey Baratheon 
                             16                               4                              19 
                     Jojen Reed                        Jon Snow                   Jorah Mormont 
                              3                              13                               3 
                   Loras Tyrell                           Luwin                      Lysa Arryn 
                              6                               6                               4 
                   Mance Rayder                 Margaery Tyrell                      Meera Reed 
                              1                               5                               3 
                     Melisandre                     Meryn Trant              Myrcella Baratheon 
                              3                               3                               3 
                  Petyr Baelish                         Pycelle                 Renly Baratheon 
                             10                               6                              10 
                   Rickon Stark                      Robb Stark                Robert Baratheon 
                              6                              17                              16 
                  Rodrik Cassel                   Samwell Tarly                  Sandor Clegane 
                              5                               5                               6 
                    Sansa Stark               Stannis Baratheon                   Theon Greyjoy 
                             17                              13                               6 
               Tommen Baratheon                Tyrion Lannister                 Tywin Lannister 
                              6                              25                               9 
                Quentyn Martell                           Varys 
                              2                               7 

which.max(degree(g))

Tyrion Lannister 
              47 

strength(g)

Aemon Targaryen (Maester Aemon)                      Arya Stark                 Barristan Selmy 
                            234                             512                             226 
                     Bran Stark                Brienne of Tarth                           Bronn 
                           1206                             225                             160 
                  Catelyn Stark                Cersei Lannister              Daenerys Targaryen 
                            766                            1438                             547 
                 Davos Seaworth                           Drogo                    Eddard Stark 
                            195                             151                            1175 
                   Edmure Tully                  Gregor Clegane                           Grenn 
                            128                             102                             121 
               Hizdahr zo Loraq                           Hodor                      Ilyn Payne 
                            138                             333                             112 
                Jaime Lannister                    Jeor Mormont               Joffrey Baratheon 
                            862                             277                            1343 
                     Jojen Reed                        Jon Snow                   Jorah Mormont 
                            223                            1238                             216 
                   Loras Tyrell                           Luwin                      Lysa Arryn 
                            167                             238                             179 
                   Mance Rayder                 Margaery Tyrell                      Meera Reed 
                            112                             258                             255 
                     Melisandre                     Meryn Trant              Myrcella Baratheon 
                            208                             102                              87 
                  Petyr Baelish                         Pycelle                 Renly Baratheon 
                            477                             193                             448 
                   Rickon Stark                      Robb Stark                Robert Baratheon 
                            263                             966                            1091 
                  Rodrik Cassel                   Samwell Tarly                  Sandor Clegane 
                            137                             472                             259 
                    Sansa Stark               Stannis Baratheon                   Theon Greyjoy 
                           1059                             771                             239 
               Tommen Baratheon                Tyrion Lannister                 Tywin Lannister 
                            337                            1694                             392 
                Quentyn Martell                           Varys 
                             80                             400 

which.max(strength(g))

Tyrion Lannister 
              47 

Neighborhood of a Node

neighbors(g, 'Robert Baratheon')

+ 16/50 vertices, named, from 17b0991:
 [1] Barristan Selmy    Catelyn Stark      Cersei Lannister   Daenerys Targaryen Eddard Stark      
 [6] Jaime Lannister    Joffrey Baratheon  Jon Snow           Petyr Baelish      Pycelle           
[11] Renly Baratheon    Sansa Stark        Stannis Baratheon  Tyrion Lannister   Tywin Lannister   
[16] Varys             

ego network

ego(g, 2, "Drogo")[[1]]

+ 8/50 vertices, named, from 17b0991:
[1] Drogo              Daenerys Targaryen Jorah Mormont      Barristan Selmy    Hizdahr zo Loraq   Robert Baratheon  
[7] Quentyn Martell    Tyrion Lannister  

g.drogo <- make_ego_graph(g, 2, nodes = "Drogo")[[1]]
g.danny <- make_ego_graph(g, 2, nodes = "Daenerys Targaryen")[[1]]

plot(g.drogo)

plot(g.danny)

Btw: To merge two graphs, just do:

g.merge

IGRAPH 45a00e4 UN-- 21 82 -- 
+ attr: name (v/c), weight_1 (e/n), weight_2 (e/n)
+ edges from 45a00e4 (vertex names):
 [1] Stannis Baratheon--Tywin Lannister   Renly Baratheon  --Stannis Baratheon Pycelle          --Varys            
 [4] Petyr Baelish    --Varys             Petyr Baelish    --Sansa Stark       Petyr Baelish    --Pycelle          
 [7] Jon Snow         --Stannis Baratheon Joffrey Baratheon--Varys             Joffrey Baratheon--Tywin Lannister  
[10] Joffrey Baratheon--Stannis Baratheon Joffrey Baratheon--Sansa Stark       Joffrey Baratheon--Renly Baratheon  
[13] Joffrey Baratheon--Petyr Baelish     Jaime Lannister  --Tywin Lannister   Jaime Lannister  --Sansa Stark      
[16] Jaime Lannister  --Renly Baratheon   Jaime Lannister  --Joffrey Baratheon Eddard Stark     --Varys            
[19] Eddard Stark     --Stannis Baratheon Eddard Stark     --Sansa Stark       Eddard Stark     --Renly Baratheon  
[22] Eddard Stark     --Pycelle           Eddard Stark     --Petyr Baelish     Eddard Stark     --Jon Snow         
+ ... omitted several edges

Betweenness Centrality

betweenness(g)

Aemon Targaryen (Maester Aemon)                      Arya Stark                 Barristan Selmy 
                            0.0                            19.0                            93.0 
                     Bran Stark                Brienne of Tarth                           Bronn 
                           17.5                             0.0                             0.0 
                  Catelyn Stark                Cersei Lannister              Daenerys Targaryen 
                           94.0                            96.5                             0.0 
                 Davos Seaworth                           Drogo                    Eddard Stark 
                            0.0                             0.0                           138.0 
                   Edmure Tully                  Gregor Clegane                           Grenn 
                            0.0                            15.0                             0.0 
               Hizdahr zo Loraq                           Hodor                      Ilyn Payne 
                            0.0                            85.0                             8.5 
                Jaime Lannister                    Jeor Mormont               Joffrey Baratheon 
                           67.5                            57.0                            63.0 
                     Jojen Reed                        Jon Snow                   Jorah Mormont 
                            0.0                           116.0                            47.5 
                   Loras Tyrell                           Luwin                      Lysa Arryn 
                           57.5                           133.0                             0.0 
                   Mance Rayder                 Margaery Tyrell                      Meera Reed 
                            0.0                             0.0                             0.0 
                     Melisandre                     Meryn Trant              Myrcella Baratheon 
                           19.0                             0.0                             1.0 
                  Petyr Baelish                         Pycelle                 Renly Baratheon 
                            7.0                            25.0                            31.0 
                   Rickon Stark                      Robb Stark                Robert Baratheon 
                           73.0                            98.5                           155.5 
                  Rodrik Cassel                   Samwell Tarly                  Sandor Clegane 
                           25.0                            20.0                            13.0 
                    Sansa Stark               Stannis Baratheon                   Theon Greyjoy 
                          137.0                           103.0                            30.0 
               Tommen Baratheon                Tyrion Lannister                 Tywin Lannister 
                            0.0                           309.0                            24.0 
                Quentyn Martell                           Varys 
                            0.0                             0.0 

Eigenvector Centrality

eigen_centrality(g, scale = TRUE)$vector %>% round(3)

Aemon Targaryen (Maester Aemon)                      Arya Stark                 Barristan Selmy 
                          0.064                           0.332                           0.073 
                     Bran Stark                Brienne of Tarth                           Bronn 
                          0.297                           0.126                           0.183 
                  Catelyn Stark                Cersei Lannister              Daenerys Targaryen 
                          0.415                           0.952                           0.039 
                 Davos Seaworth                           Drogo                    Eddard Stark 
                          0.060                           0.007                           0.754 
                   Edmure Tully                  Gregor Clegane                           Grenn 
                          0.067                           0.072                           0.035 
               Hizdahr zo Loraq                           Hodor                      Ilyn Payne 
                          0.008                           0.079                           0.086 
                Jaime Lannister                    Jeor Mormont               Joffrey Baratheon 
                          0.547                           0.113                           0.922 
                     Jojen Reed                        Jon Snow                   Jorah Mormont 
                          0.046                           0.364                           0.038 
                   Loras Tyrell                           Luwin                      Lysa Arryn 
                          0.112                           0.061                           0.121 
                   Mance Rayder                 Margaery Tyrell                      Meera Reed 
                          0.047                           0.200                           0.051 
                     Melisandre                     Meryn Trant              Myrcella Baratheon 
                          0.065                           0.089                           0.067 
                  Petyr Baelish                         Pycelle                 Renly Baratheon 
                          0.353                           0.156                           0.267 
                   Rickon Stark                      Robb Stark                Robert Baratheon 
                          0.098                           0.428                           0.755 
                  Rodrik Cassel                   Samwell Tarly                  Sandor Clegane 
                          0.044                           0.127                           0.186 
                    Sansa Stark               Stannis Baratheon                   Theon Greyjoy 
                          0.723                           0.367                           0.091 
               Tommen Baratheon                Tyrion Lannister                 Tywin Lannister 
                          0.270                           1.000                           0.333 
                Quentyn Martell                           Varys 
                          0.004                           0.345 

graph structure

edge_density(g)

[1] 0.1542857

diameter(g, directed = F, weights = NA)

[1] 4

transitivity(g)

[1] 0.4552807

mean_distance(g, directed = F)

[1] 2.337959

Your turn

Ahh, you saw it comming, right? What about you explore the GoT network a bit on your own HERE. Lets see how that works out!

Directed Networks are comming…

So far so good, up to now we considered undirected networks, constructed by the amount characters co-occur. However, as you already might guess, that’s not where we stop.

There are also other relationships to which we can, and sometimes have to, assign a directionality. An obvious example here are family ties. Here, I will ose the nicely compiled dataset of the wonderful Shirin that can be found here. It contains a nodelist with house-affiliations and furtehr characteristics of main characters, and a edgelist of their family relationships.

rm(chars.main, g, g.danny, g.drogo, g.merge)
edges.fam <- readRDS("../data/GoT/union_edges.RDS")
nodes.fam <- readRDS("../data/GoT/union_characters.RDS")

head(nodes.fam)

head(edges.fam)

g <- graph_from_data_frame(edges.fam, 
                           vertices = nodes.fam,
                           directed = TRUE)
g

IGRAPH 813c92b DN-- 208 404 -- 
+ attr: name (v/c), male (v/n), culture (v/c), house (v/c), popularity (v/n), house2 (v/c), color (v/c),
| shape (v/c), type (e/c), color (e/c), lty (e/c)
+ edges from 813c92b (vertex names):
 [1] Lysa Arryn       ->Robert Arryn       Jasper Arryn     ->Alys Arryn        
 [3] Jasper Arryn     ->Jon Arryn          Jon Arryn        ->Robert Arryn      
 [5] Cersei Lannister ->Tommen Baratheon   Cersei Lannister ->Joffrey Baratheon 
 [7] Cassana Baratheon->Stannis Baratheon  Cersei Lannister ->Myrcella Baratheon
 [9] Selyse Florent   ->Shireen Baratheon  Cassana Baratheon->Renly Baratheon   
[11] Rhaelle Targaryen->Steffon Baratheon  Cassana Baratheon->Robert Baratheon  
[13] Robert Baratheon ->Tommen Baratheon   Robert Baratheon ->Joffrey Baratheon 
+ ... omitted several edges

plot(g)

For plotting the legend, I am summarizing the edge and node colors.

color_vertices <- nodes.fam %>%
  group_by(house, color) %>%
  summarise(n = n()) %>%
  filter(!is.na(color))
colors_edges <- edges.fam %>%
  group_by(type, color) %>%
  summarise(n = n()) %>%
  filter(!is.na(color))

plot(g,
     layout = layout_with_fr(g),
     vertex.label = gsub(" ", "\n", V(g)$name),
     vertex.shape = V(g)$shape,
     vertex.color = V(g)$color, 
     vertex.size = (V(g)$popularity + 0.5) * 5, 
     vertex.frame.color = "gray", 
     vertex.label.color = "black", 
     vertex.label.cex = 0.8,
     edge.arrow.size = 0.5,
     edge.color = E(g)$color,
     edge.lty = E(g)$lty)
legend("topleft", legend = c(NA, "Node color:", as.character(color_vertices$house), NA, "Edge color:", as.character(colors_edges$type)), pch = 10,
       col = c(NA, NA, color_vertices$color, NA, NA, colors_edges$color), pt.cex = 3, cex = 2, bty = "n", ncol = 1,
       title = "") 

legend("topleft", legend = "", cex = 3, bty = "n", ncol = 1,
       title = "Game of Thrones Family Ties")

Community Detection

You might have already guessed, we can very well also do a clustering exercise in networks. Here, we do not cluster nodes according to their similarity in attributes, but according to their connectivity. There are plenty of algorithms,, and we will explore further ones lateron. Most of them aim to find communities with maximum within/connectivity, and minimum between/connectivity.

However, most of them are not designed to work with directed networks. Therefore, we will convert our nice network for now to an undirected one.

g.ud <- as.undirected(g)

First, we will give it a try with the edge-betweenness algorithm (Newman-Girvan). Here, high-betweenness edges are removed sequentially (recalculating at each step) and the best partitioning of the network is selected.

Lets take a look how it works.

And we now run it. As an hirarchical community detection technique. Since it is an hirarchical one, we can again plot a dendogram which we already know from the hirarchical clustering

ceb

IGRAPH clustering edge betweenness, groups: 12, mod: 0.84
+ groups:
  $`1`
   [1] "Alys Arryn"       "Elys Waynwood"    "Jasper Arryn"     "Jeyne Royce"      "Jon Arryn"       
   [6] "Lysa Arryn"       "Robert Arryn"     "Rowena Arryn"     "Edmure Tully"     "Sansa Stark"     
  [11] "Arya Stark"       "Bran Stark"       "Catelyn Stark"    "Eddard Stark"     "Jeyne Westerling"
  [16] "Rickon Stark"     "Robb Stark"       "Talisa Stark"     "Hoster Tully"     "Minisa Whent"    
  [21] "Petyr Baelish"   
  
  $`2`
   [1] "Cassana Baratheon"  "Cersei Lannister"   "Jaime Lannister"    "Joffrey Baratheon"  "Margaery Tyrell"   
   [6] "Myrcella Baratheon" "Renly Baratheon"    "Robert Baratheon"   "Selyse Florent"     "Shireen Baratheon" 
  + ... omitted several groups/vertices

plot(ceb, g.ud,
     vertex.frame.color = V(g)$color, # load the predefined color of the nodes (houses)
     vertex.size = (V(g)$popularity + 0.5) * 5 # define node-size by popularity) 
)

Lets only see how good it performs on the major houses, the rest is too small anyhow

bind_cols(com = ceb$membership, house = V(g.ud)$house) %>%
  group_by(house) %>%
  filter(n() >= 10) %>%
  ungroup() %>%
  table()

    house
com  House Baratheon House Frey House Greyjoy House Lannister House Martell House Stark House Targaryen House Tyrell
  1                0          0             0               0             0           7               0            0
  2                9          0             0               4             0           0               0            1
  3                1          0             0               0             2           0              13            0
  4                0         13             0               0             0           0               0            0
  5                0          7             0               9             0           0               0            0
  6                0          0            14               0             0           0               0            0
  7                0          0             0              13             0           0               0            0
  8                0          0             0               0            11           0               0            0
  9                0          0             0               0             0           6               0            0
  10               0          0             0               0             0          13               0            0
  11               0          0             0               0             0           0               0            9
  12               0          0             0               0             0           0               0            5

We see, indeed, that the communities for the most part capture the affiliation to the great houses.

Your turn

AGain, its time to have some fun on your own. HERE you will find another kaggle notebook where you can demonstrate your network analysis skills even more!